We report a simple, versatile in situ transmission electron microscopy (TEM) approach for investigating the nucleation and growth mechanism of carbon nanotubes (CNTs), by which the composition, phase transition and physical state of various catalysts can be clearly resolved. In our approach, catalyst nanoparticles (NPs) are placed in a multi-wall CNT "tubular furnace" with two open ends, and a high temperature is obtained by Joule heating in the specimen chamber of a TEM. The carbon is supplied by electron irradiation-induced injection of carbon atoms. Comparative studies on the catalytic behavior of traditional iron oxide and recently discovered gold catalysts were performed. It was found that the growth of CNTs from iron oxide involves the reduction of Fe2O3 to Fe3C, nucleation and growth of CNTs from partially liquefied Fe3C, and finally the formation of metallic Fe when the growth stops. In contrast, while changes in shape, size and orientation were also observed for the fluctuating Au NPs, no chemical reactions or phase transitions occurred during the nucleation of CNTs. These two distinct nucleation and growth processes and mechanisms would be valuable for the structure-controlled growth of CNTs by catalyst design and engineering.